The Bayes factor is the ratio of the marginal likelihoods
under two different models (see Kass & Raftery, 1995). Function
`varbvsbf`

provides a convenient interface for computing the
Bayes factor comparing the fit of two different `varbvs`

models.

varbvsbf (fit0, fit1)
bayesfactor (logw0, logw1)

## Arguments

fit0 |
An output returned from `varbvs` . |

fit1 |
Another output returned from `varbvs` . |

logw0 |
log-probabilities or log-importance weights under H0. |

logw1 |
log-probabilities or log-importance weights under H1. |

## Value

The estimated Bayes factor.

## Details

Computes numerical estimate of
$$
BF = Pr(data | H1) / Pr(data | H0),
$$
the probability of the data given the "alternative" hypothesis (H1) over
the probability of the data given the "null" hypothesis (H0). This is
also known as a Bayes factor (see Kass & Raftery, 1995). Here we assume
that although these probabilities cannot be computed analytically
because they involve intractable integrals, we can obtain reasonable
estimates of these probabilities with a simple numerical approximation
over some latent variable assuming the prior over this latent variable
is uniform. The inputs are the log-probabilities
$$
Pr(data, Z0 | H0) = Pr(data | Z0, H0) x Pr(Z0 | H0),
Pr(data, Z1 | H1) = Pr(data | Z1, H1) x Pr(Z1 | H1),
$$
where Pr(Z0 | H0) and Pr(Z1 | H1) are uniform over all Z0 and Z1.

Alternatively, this function can be viewed as computing an importance
sampling estimate of the Bayes factor; see, for example, R. M. Neal,
"Annealed importance sampling", Statistics and Computing, 2001. This
formulation described above is a special case of importance sampling
when the settings of the latent variable Z0 and A1 are drawn from the
same (uniform) distribution as the prior, Pr(Z0 | H0) and Pr(Z1 | H1),
respectively.

## References

P. Carbonetto and M. Stephens (2012). Scalable variational inference
for Bayesian variable selection in regression, and its accuracy in
genetic association studies. *Bayesian Analysis* **7**,
73--108.

R. E. Kass and A. E. Raftery (1995). Bayes Factors. *Journal of the
American Statistical Association* **90**, 773--795.

R. M. Neal (2001). Annealed importance sampling. *Statistics and
Computing* **11**, 125--139.

## See also

`varbvs`

, `normalizelogweights`